A090861 -id:A090861 - cp's OEIS frontend

A217010 Permutation of natural numbers arising from applying the walk of a square spiral (e.g., A214526) to the data of right triangular type-1 spiral (defined in A214230).

1, 2, 13, 3, 5, 6, 7, 8, 9, 10, 12, 32, 61, 33, 14, 4, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 11, 31, 60, 98, 145, 99, 62, 34, 15, 17, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 28, 30, 59, 97, 144, 200, 265, 201, 146, 100, 63, 35, 16, 39, 71

Offset: 1

Alex Ratushnyak, Sep 23 2012

			Triangular spiral (A214230) begins:
.
  56
   | \
  55  57
   |     \
  54  29  58
   |   | \   \
  53  28  30  59
   |   |     \   \
  52  27  11  31  60
   |   |   | \   \   \
  51  26  10  12  32  61
   |   |   |     \   \   \
  50  25   9   2  13  33  62
   |   |   |   | \   \   \   \
  49  24   8   1   3  14  34  63
   |   |   |         \   \   \   \
  48  23   7---6---5---4  15  35  64
   |   |                     \   \   \
  47  22--21--20--19--18--17--16  36  65
   |                                 \   \
  46--45--44--43--42--41--40--39--38--37  66
                                             \
  78--77--76--75--74--73--72--71--70--69--68--67
.
Square spiral (defining order in which elements are fetched) begins:
.
  49  26--27--28--29--30--31
   |   |                   |
  48  25  10--11--12--13  32
   |   |   |           |   |
  47  24   9   2---3  14  33
   |   |   |   |   |   |   |
  46  23   8   1   4  15  34
   |   |   |       |   |   |
  45  22   7---6---5  16  35
   |   |               |   |
  44  21--20--19--18--17  36
   |                       |
  43--42--41--40--39--38--37

Cf. A090861, A214526, A214230.

SIZE = 33       # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk(0, -1,  1,  1)    # up
    walk( 1, 1, -1,  0)    # right-down
    if posX==SIZE-1:
        break
    walk(-1, 0,  0, -1)    # left
import sys
grid2 = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid2[posY*SIZE+posX]=1
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        sys.exit(1)
    print(a, end=', ')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX:
    walk2(0, -1, 1, 0)    # up
    walk2(1, 0, 0, 1)     # right
    walk2(0, 1, -1, 0)    # down
    walk2(-1, 0, 0, -1)   # left

A090915 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 8, 7, 6, 5, 4, 3, 2, 9, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 25, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 49, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a square clockwise spiral: 1 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 2 -> 9 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090925, A090928, A090929, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n==(2*x+1)^2, n, 8*x^2 -n+2], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n == (2*s(n)+1)^2, n, 8*s(n)^2-n+2), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n if n == (2*x+1)^2 else 8*x^2 + 2 - n
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A090925 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 9, 2, 3, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 10, 11, 12, 13, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a square counterclockwise spiral: 1 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 2 -> 3 -> 14 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090915, A090928, A090929, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n+2*x <= (2*x+1)^2, n +2*x, n-6*x], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n+2*s(n) <= (2*s(n)+1)^2, n +2*s(n), n - 6*s(n)), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n + 2*x if n + 2*x <= (2*x+1)^2 else n - 6*x
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A090928 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 6, 7, 8, 9, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 24, 25, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 50, 51, 52, 53, 54, 55, 56

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a counterclockwise spiral: 1 -> 6 -> 7 -> 8 -> 9 -> 2 -> 3 -> 4 -> 5 -> 18 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers

Cf. A020703, A090861, A090915, A090925, A090929, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]]+1)/2]}, Table[If[n +4*x <= (2*x+1)^2, n+4*x, n-4*x], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n+4*s(n) <= (2*s(n)+1)^2, n +4*s(n), n - 4*s(n)), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n + 4*x if n + 4*x <= (2*x+1)^2 else n - 4*x
[a(n) for n in (1..75)] # Eric M. Schmidt, May 18 2016

a(n) = A090925(A090925(n)). - Rémy Sigrist, Jul 25 2025

Offset corrected by Eric M. Schmidt, May 18 2016

A090929 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 8, 9, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 74, 75, 76, 77, 78, 79, 80, 81, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a counterclockwise spiral: 1 -> 8 -> 9 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 22 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090915, A090925, A090928, A090930.

With[{x = Floor[(Floor[Sqrt[n-1]] +1)/2]}, Table[If[n +6*x <= (2*x+1)^2, n +6*x, n -2*x], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(if(n+6*s(n) <= (2*s(n)+1)^2, n +6*s(n), n - 2*s(n)), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return n + 6*x if n + 6*x <= (2*x+1)^2 else n - 2*x
[a(n) for n in (1..75)]
# Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A090930 Permutation of natural numbers arising from a square spiral.

Original entry on oeis.org

1, 2, 9, 8, 7, 6, 5, 4, 3, 12, 11, 10, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 30, 29, 28, 27, 26, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 56, 55, 54, 53, 52, 51, 50, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66

Offset: 1

Felix Tubiana, Feb 26 2004

Write out the natural numbers in a square counterclockwise spiral:
.
  17--16--15--14--13
   |               |
  18   5---4---3  12
   |   |       |   |
  19   6   1---2  11
   |   |           |
  20   7---8---9--10
   |
  21--22--23--24--25
.
Now read off the numbers in a clockwise spiral: 1 -> 2 -> 9 -> 8 -> 7 -> 6 -> 5 -> 4 -> 3 -> 12 -> etc.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A020703, A090861, A090915, A090925, A090928, A090929.

With[{x = Floor[(Floor[Sqrt[n-1]] +1)/2]}, Table[8*x^2 -n +2 +x*If[n <= 4*x^2 -2*x, -6, 2], {n, 1, 75}]] (* G. C. Greubel, Feb 05 2019 *)

{s(n) = ((sqrtint(n-1)+1)/2)\1};
for(n=1,75, print1(8*s(n)^2 -n +2 +s(n)*if(n <= 2*s(n)*(2*s(n)-1), -6, 2), ", ")) \\ G. C. Greubel, Feb 05 2019

def a(n):
    x = (isqrt(n-1)+1)//2
    return 8*x^2 + (-6 if n <= 4*x^2 - 2*x else 2)*x + 2 - n
[a(n) for n in (1..75)]
# Eric M. Schmidt, May 18 2016

Offset corrected by Eric M. Schmidt, May 18 2016

A217015 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of rotated-square spiral (defined in A215468).

Original entry on oeis.org

1, 5, 6, 2, 8, 3, 10, 4, 12, 24, 13, 14, 27, 15, 7, 17, 31, 18, 9, 20, 35, 21, 11, 23, 39, 59, 40, 25, 26, 43, 64, 44, 28, 16, 30, 48, 70, 49, 32, 19, 34, 53, 76, 54, 36, 22, 38, 58, 82, 110, 83, 60, 41, 42, 63, 88, 117, 89, 65, 45, 29, 47, 69, 95, 125, 96, 71, 50

Offset: 1

Alex Ratushnyak, Sep 23 2012

Cf. A090861, A214526, A215468, A217010.

SIZE = 33       # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
posX += 1
grid[posY*SIZE+posX]=2
n = 3
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX!=SIZE-1:
    walk(-1,  1, -1, -1)    # down-left
    walk(-1, -1,  1, -1)    # up-left
    walk( 1, -1,  1,  0)    # up-right
    walk( 1,  0,  1,  1)    # right
    walk( 1,  1, -1,  1)    # down-right
import sys
grid2 = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid2[posY*SIZE+posX]=1
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        sys.exit(1)
    print(a, end=', ')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk2(0, -1, 1, 0)    # up
    walk2(1, 0, 0, 1)     # right
    walk2(0, 1, -1, 0)    # down
    walk2(-1, 0, 0, -1)   # left

A217011 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of right triangular type-2 spiral (defined in A214251).

Original entry on oeis.org

1, 5, 19, 6, 8, 9, 10, 2, 3, 4, 18, 41, 73, 42, 20, 7, 24, 25, 26, 27, 28, 11, 12, 13, 14, 15, 17, 40, 72, 113, 163, 114, 74, 43, 21, 23, 49, 50, 51, 52, 53, 54, 55, 29, 30, 31, 32, 33, 34, 35, 16, 39, 71, 112, 162, 221

Offset: 1

Alex Ratushnyak, Sep 23 2012

Cf. A090861, A214526, A214251, A217010.

SIZE = 33       # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk(-1, 0,  0, -1)    # left
    walk(0, -1,  1,  1)    # up
    if posY==0:
        break
    walk( 1, 1, -1,  0)    # right-down
import sys
grid2 = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid2[posY*SIZE+posX]=1
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        sys.exit(1)
    print(a, end=', ')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk2(0, -1, 1, 0)    # up
    walk2(1, 0, 0, 1)     # right
    walk2(0, 1, -1, 0)    # down
    walk2(-1, 0, 0, -1)   # left

A217012 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of right triangular type-3 spiral (defined in A214252).

Original entry on oeis.org

1, 8, 25, 9, 2, 3, 4, 5, 6, 7, 24, 50, 85, 51, 26, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 49, 84, 128, 181, 129, 86, 52, 27, 11, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 22, 48, 83, 127, 180, 242

Offset: 1

Alex Ratushnyak, Sep 23 2012

Cf. A090861, A214526, A214252, A217010.

SIZE = 33       # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posY!=0:
    walk( 1, 1, -1,  0)    # right-down
    walk(-1, 0,  0, -1)    # left
    walk(0, -1,  1,  1)    # up
import sys
grid2 = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid2[posY*SIZE+posX]=1
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        sys.exit(1)
    print(a, end=', ')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk2(0, -1, 1, 0)    # up
    walk2(1, 0, 0, 1)     # right
    walk2(0, 1, -1, 0)    # down
    walk2(-1, 0, 0, -1)   # left

A217013 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of triangular horizontal-first spiral (defined in A214250).

Original entry on oeis.org

1, 3, 14, 4, 6, 7, 8, 2, 12, 30, 13, 32, 59, 33, 15, 5, 19, 20, 21, 22, 23, 9, 11, 29, 55, 89, 56, 31, 58, 93, 136, 94, 60, 34, 16, 18, 40, 41, 42, 43, 44, 45, 46, 24, 10, 28, 54, 88, 130, 180, 131, 90, 57, 92, 135, 186, 245, 187, 137, 95, 61, 35, 17, 39, 69

Offset: 1

Alex Ratushnyak, Sep 23 2012

Cf. A090861, A214526, A214250, A217010.

SIZE = 33       # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX

cp's OEIS Frontend

A217010 Permutation of natural numbers arising from applying the walk of a square spiral (e.g., A214526) to the data of right triangular type-1 spiral (defined in A214230).

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A090915 Permutation of natural numbers arising from a square spiral.

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A090925 Permutation of natural numbers arising from a square spiral.

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A090928 Permutation of natural numbers arising from a square spiral.

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A090929 Permutation of natural numbers arising from a square spiral.

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Mathematica

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Sage

Extensions

A090930 Permutation of natural numbers arising from a square spiral.

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Mathematica

PARI

Sage

Extensions

A217015 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of rotated-square spiral (defined in A215468).

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Programs

Python

A217011 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of right triangular type-2 spiral (defined in A214251).

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